There are many kinds of adders and subtractors known in the prior art. Some are analog devices, but for handling many digits with speed and accuracy, the primary focus has turned to digital signal-processing devices. Many kinds of digital adders and subtractors have been explored in the prior art, as described for example in Hill, Fredrick J. and Peterson, Gerald R., Introduction to Switching Theory and Logic Design, 3rd ed. (Wiley 1981).
Those known digital adders and subtractors rely on sequential processing. That is, integers being added and subtracted are digitalized into representations having a number of bits in bit positions. For each bit position there are levels of gaging, whereby a result computed by each bit position device is fed into the next bit position device in a cascade of devices: "The addition process may thus be carried out one digit at a time, starting with the least significant digit." Switching Theory at page 176. (This gives rise to the notion of a "carry" propagation from one bit position to the next--a signal output from the device computing the first bit to the device computing the next bit position.)
A consequence of such prior art designs processing is slowed by the levels of gaging and the sequence of processing used to add and subtract integers represented with more than one bit. In applications requiring speed, the slowness of the sequential processing is a serious consequence.
Also, according to Switching Theory at pages 180-181, "there are, unfortunately, no simple rules or completely specified procedures for finding such designs." Thus, it is difficult to design device, in the absence of a generalized method.
Further, the adders and subtractors of the known prior art must each find a way to deal with the respective signs of the integers. For example, the sign at the first number can be treated as positive and the sign at the second number can be used in specifying that there is addition or subtraction of the two numbers. Often the handling of the signs of the respective integers limits the flexibility of the device and adds delay to the processing.
Additionally, a design in the known prior art that can be implemented in one media, say, electrical, may be unworkably difficult to implement in another media, say, optical. Limited flexibility is another problem in the prior art.